Tesseract
|symmetry = Full hexadecachoric symmetry |vertex_count = 16 |edges_length = 32l |surface_area = 24l^2 |surcell_volume = 8l^3 |surteron_bulk = l^4 |vertices = 16 points |edges = 32 line segments |faces = 24 squares |cells = 8 cubes |tera = 1 tesseract |image1 = Tesseract.jpg |caption1=A cell-on 3-D projection of a tesseract.}} A tesseract or octachoron is a fourth-dimensional hypercube. Since the number of dimensions is a square number, the diagonal length of a tesseract is an integer - in this case, 2. Its Bowers acronym is "tes". It is one of the three regular polychora that can tile 4-dimensional space, forming the tesseractic tetracomb. Under the elemental naming scheme it is called a geochoron. Tesseract Rubik's cubes can found online, but cannot be built in the 3D world. Hypercube Products The tesseract can be expressed as a hypercube product, potentially with less symmetry than the uniform and regular ideal tesseract, in five different ways: \{4,3,3\} - tesseract As a tesseract, the hypervolumes can be expressed in terms of a single variable, the edge length l. This is the most symmetrical variant of the tesseract. \{4,3\} \times \{\} - cube prism As a cube prism, the hypervolumes require two lengths to express: the edge length a of the cube, and the height b of the prism. * edge length = 8\left(3a + b\right) * surface area = 12a \left( a + b \right) * surcell volume = 2a^2 \left( a + 3b \right) * surteron bulk = a^3 b When a=b, this becomes the symmetrical tesseract. \{4\} \times \{\}^2 - square prism prism As a square prism prism, the hypervolumes require three lengths to express: the edge length a of the square, and the seperate heights b and c of the two prisms. * edge length = 8 \left( 2a + b + c \right) * surface area = 4 \left( a^2 + 2 ab + 2 ac + bc \right) * surcell volume = 2a \left( ab + ac + 2bc \right) * surteron bulk = a^2 b c When a=b xor a=c, this becomes the cubic prism. When b=c, this becomes the square duoprism. When a=b=c, this becomes the symmetrical tesseract. \{\}^4 - line prism prism prism As a line prism prism prism, the hypervolumes require four lengths to express. This is the least symmetrical variant of the tesseract. * edge length = 8\left( a + b + c + d \right) * surface area = 4 \left( ab + ac + ad + bc + bd + cd \right) * surcell volume = 2 \left( abc + abd + acd + bcd \right) * surteron bulk = abcd When a=b and c=d, a=c and b=d, xor a=d and b=c, this becomes the square duoprism. When a=b=c, b=c=d, a=c=d xor a=b=d, this becomes the cubic prism. When a=b, a=c, a=d, b=c, b=d xor c=d, this becomes the square prism prism. When a=b=c=d, this becomes the symmetrical tesseract. \{4\}^2 - square duoprism As a square duoprism, the hypervolumes require two lengths to express: the seperate edge lengths a and b of the two squares. * edge length = 16 \left( a + b \right) * surface area = 4 \left( a^2 + 4 ab + b^2 \right) * surcell volume = 4ab \left( a + b \right) * surteron bulk = a^2 b^2 When a=b, this becomes the symmetrical tesseract. Properties The tesseract can be exactly decomposed into eight cubic pyramids with unit side length. This is because the distance between a vertex and the center is the same as the edge length. If these pyramids are joined to the cubes of the tesseract the result is the icositetrachoron - the square pyramidal cells merge into octahedra. Symbols Dynkin symbols of the tesseract include: *x4o3o3o (regular) *x x4o3o (cubic prism) *x4o x4o (square duoprism) *x x x4o (square diprism) *x x x x (tesseractic block) *xx4oo3oo&#xt, xx xx4oo&#xt, xx xx xx&#xt (as cube atop cube) *oqooo3ooqoo3oooqo&#xt (vertex first) *xxx4ooo oqo&#xt, xxx xxx oqo&#xt (square first) *xxxx oqoo3ooqo&#xt (edge first) *qo3oo3oq *b3oo&#zx (sum of two demitesseracts) *xx xx qo oq&#zx (rhombic diprism) *xx qo3oo3oq&#zx *prism of sum of two tetrahedra) Structure and Sections Structure The tesseract is composed of 8 cubic cells. Two of these cubes line in parallel 3-D spaces, while the remaining six connect the faces of the cubes. Four cubes meet at each vertex. In cube-first position, it is a sequence of identical cubes. In square-centered orientation, it is a square which expands to a square prism and back. When seen line-first it is a line that expands to a triangular prism, then turns to a hexagonal prism, and then back. Finally in corner first orientation, it goes through the entire tetrahedral truncation series, from point to tetrahedron to octahedron in the middle and then back. Hypervolumes * vertex count = 16 * edge length = 32l * surface area = 24l^2 * surcell volume = 8l^3 * surteron bulk = l^4 Subfacets * 16 points (0D) * 32 line segments (1D) * 24 squares (2D) * 8 cubes (3D) * 1 tesseract (4D) Radii *Vertex radius: l *Edge radius: \frac{\sqrt{3}}{2}l *Face radius: \frac{\sqrt{2}}{2}l *Cell radius: 1/2l Angles *Dichoral angle: 90º Vertex coordinates The vertices of a tesseract with side 2 can be denoted on a 4D Cartesian plane by (±1,±1,±1,±1). Equations The surface of a tesseract can be graphed by the equation max(x^2,y^2,z^2,w^2) = 1 Notations *Toratopic notation: |||| *Tapertopic notation: 1111 Related shapes *Dual: Hexadecachoron *Vertex figure: Tetrahedron, edge length \sqrt{2} See Also Category:Shape Category:4 dimensional Category:Hypercubes Category:Rotatopes Category:Polychora Category:Regular polychora Category:Uniform polychora Category:Toratopes Category:Tapertopes